From formal errors towards realistic uncertainties

    Evaluation of uncertainties of geodetic parameter estimates 
is the problem that is not yet solved in a satisfactory way. 
A direct evaluation of the uncertainties derived from a least 
square solution is labeled "formal" and is usually biased, 
sometimes up to an order of magnitude. Customary, the use of 
formal errors for scientific analysis is discouraged. We claim 
that the root of the problem is neglecting off-diagonal elements 
in the variance-covariance matrix of the noise in the data. 
A careful reconstruction of the full variance-covariance matrix, 
including the off-diagonal terms greatly improves realism of 
uncertainty estimates derived from least squares. We processed 
the dataset of VLBI group delays and built the a priori 
variance-covariance of the atmosphere-driven noise based on 
analysis of the output of NASA high-resolution numerical weather 
models. We found that the uncertainties of parameter estimates 
derived from this least square solution that uses such 
variance-covariance matrices become much closer to realistic 
errors. We consider approaches for for implementation of this 
method in routine data analysis of space geodesy data.